https://github.com/cran/sn
Tip revision: 4643febbf85e54a5f87fee8800e183afb48bfdb1 authored by Adelchi Azzalini on 10 February 2017, 09:53:49 UTC
version 1.5-0
version 1.5-0
Tip revision: 4643feb
dst.Rd
% file sn/man/dst.Rd
% This file is a component of the package 'sn' for R
% copyright (C) 2002-2014 Adelchi Azzalini
%---------------------
\name{dst}
\alias{dst}
\alias{pst}
\alias{qst}
\alias{rst}
\title{Skew-\eqn{t} Distribution}
\description{Density function, distribution function, quantiles and
random number generation for the skew-\eqn{t} (ST) distribution}
\usage{
dst(x, xi=0, omega=1, alpha=0, nu=Inf, dp=NULL, log=FALSE)
pst(x, xi=0, omega=1, alpha=0, nu=Inf, dp=NULL, method=0, ...)
qst(p, xi=0, omega=1, alpha=0, nu=Inf, tol=1e-08, dp=NULL, method=0, ...)
rst(n=1, xi=0, omega=1, alpha=0, nu=Inf, dp=NULL)
}
\arguments{
\item{x}{vector of quantiles. Missing values (\code{NA}s) are allowed.}
\item{p}{vector of probabililities.}
\item{xi}{vector of location parameters.}
\item{omega}{vector of scale parameters; must be positive.}
\item{alpha}{vector of slant parameters. With \code{pst} and \code{qst},
it must be of length 1.}
\item{nu}{a single positive value representing the degrees of freedom;
it can be non-integer. Default value is \code{nu=Inf} which corresponds
to the skew-normal distribution.
}
\item{dp}{a vector of length 4, whose elements represent location, scale
(positive), slant and degrees of freedom, respectively. If \code{dp} is
specified, the individual parameters cannot be set.}
\item{n}{a positive integer representing the sample size.}
\item{log}{logical; if TRUE, densities are given as log-densities}
\item{tol}{
a scalar value which regulates the accuracy of the result of
\code{qsn}, measured on the probability scale.
}
\item{method}{an integer value between \code{0} and \code{4} which selects
the computing method; see \sQuote{Details} below for the meaning of these
values. If \code{method=0} (default value), an automatic choice is made
among the four actual computing methods, which depends on the other
arguments.}
\item{...}{additional parameters passed to \code{integrate} or \code{pmst}.}
}
\value{Density (\code{dst}), probability (\code{pst}), quantiles (\code{qst})
and random sample (\code{rst}) from the skew-\eqn{t} distribution with given
\code{xi}, \code{omega}, \code{alpha} and \code{nu} parameters.}
\section{Details}{
Typical usages are
\preformatted{%
dst(x, xi=0, omega=1, alpha=0, nu=Inf, log=FALSE)
dst(x, dp=, log=FALSE)
pst(x, xi=0, omega=1, alpha=0, nu=Inf, method=0, ...)
pst(x, dp=, log=FALSE)
qst(p, xi=0, omega=1, alpha=0, nu=Inf, tol=1e-8, method=0, ...)
qst(x, dp=, log=FALSE)
rst(n=1, xi=0, omega=1, alpha=0, nu=Inf)
rst(x, dp=, log=FALSE)
}
}
\section{Background}{
The family of skew-\eqn{t} distributions is an extension of the Student's
\eqn{t} family, via the introduction of a \code{alpha} parameter which
regulates skewness; when \code{alpha=0}, the skew-\eqn{t} distribution
reduces to the usual Student's \eqn{t} distribution.
When \code{nu=Inf}, it reduces to the skew-normal distribution.
When \code{nu=1}, it reduces to a form of skew-Cauchy distribution.
See Chapter 4 of Azzalini & Capitanio (2014) for additional information.
A multivariate version of the distribution exists; see \code{dmst}.
}
\section{Details}{
For evaluation of \code{pst}, and so indirectly of
\code{qst}, four different methods are employed.
Method 1 consists in using \code{pmst} with dimension \code{d=1}.
Method 2 applies \code{integrate} to the density function \code{dst}.
Method 3 again uses \code{integrate} too but with a different integrand,
as given in Section 4.2 of Azzalini & Capitanio (2003), full version of
the paper.
Method 4 consists in the recursive procedure of Jamalizadeh, Khosravi and
Balakrishnan (2009), which is recalled in Complement 4.3 on
Azzalini & Capitanio (2014); the recursion over \code{nu} starts from
the explicit expression for \code{nu=1} given by \code{psc}.
Of these, Method 1 and 4 are only suitable for integer values of \code{nu}.
Method 4 becomes progressively less efficient as \code{nu} increases,
because its value corresponds to the number of nested calls, but the
decay of efficiency is slower for larger values of \code{length(x)}.
If the default argument value \code{method=0} is retained, an automatic choice
among the above four methods is made, which depends on the values of
\code{nu, alpha, length(x)}. The numerical accuracy of methods 1, 2 and 3 can
be regulated via the \code{...} argument, while method 4 is conceptually exact,
up to machine precision.
If \code{qst} is called with \code{nu>1e4}, computation is transferred to
\code{qsn}.
}
\references{
Azzalini, A. and Capitanio, A. (2003).
Distributions generated by perturbation of symmetry
with emphasis on a multivariate skew-\emph{t} distribution.
\emph{J.Roy. Statist. Soc. B} \bold{65}, 367--389.
Full version of the paper at \url{http://arXiv.org/abs/0911.2342}.
Azzalini, A. with the collaboration of Capitanio, A. (2014).
\emph{The Skew-normal and Related Families}.
Cambridge University Press, IMS Monographs series.
Jamalizadeh, A., Khosravi, M., and Balakrishnan, N. (2009).
Recurrence relations for distributions of a skew-$t$ and a linear
combination of order statistics from a bivariate-$t$.
\emph{Comp. Statist. Data An.} \bold{53}, 847--852.
}
\seealso{\code{\link{dmst}}, \code{\link{dsn}}, \code{\link{dsc}}}
\examples{
pdf <- dst(seq(-4, 4, by=0.1), alpha=3, nu=5)
rnd <- rst(100, 5, 2, -5, 8)
q <- qst(c(0.25, 0.50, 0.75), alpha=3, nu=5)
pst(q, alpha=3, nu=5) # must give back c(0.25, 0.50, 0.75)
#
p1 <- pst(x=seq(-3,3, by=1), dp=c(0,1,pi, 3.5))
p2 <- pst(x=seq(-3,3, by=1), dp=c(0,1,pi, 3.5), method=2, rel.tol=1e-9)
}
\keyword{distribution}